We apply an extended contraction principle and superexponential convergence in probability to show that a functional large deviation principle for a sequence of stochastic processes implies a corresponding functional large deviation principle for an associated sequence of first-passage-time or inverse processes. Large deviation principles are established for both inverse processes and centered inverse processes, based on corresponding results for the original process. We apply these results to obtain functional large deviation principles for renewal processes and superpositions of independent renewal processes.
"Functional large deviation principles for first-passage-time processes." Ann. Appl. Probab. 7 (2) 362 - 381, May 1997. https://doi.org/10.1214/aoap/1034625336